Inserting Juno into Orbit around Jupiter

Juno at Jupiter
Figure 1: Simulated view of Juno’s main engine firing during the craft’s insertion into orbit about Jupiter, on July 4, 2016.  Screenshot is from JPL’s Eyes on the Solar System application.

After a 5-year interplanetary journey, which included a gravity assist from Earth, NASA’s spacecraft Juno was successfully inserted into orbit about  Jupiter on July 4, 2016.  Juno is NASA’s most remote spacecraft to be entirely solar-powered; its three 8.9 meter long solar panels are the largest of any on NASA’s deep-space probes.  It houses 9 primary instruments enabling a close-up study of the planet’s gravitational field, its massive magnetic field and auroras, the chemical composition (including water) of its atmosphere, and providing detailed color images.  During the next 20 months, Juno will slip under the planet’s intense radiation belts, approaching within 5000 km of its cloud cover, and execute approximately 40 tight orbits of period 14 days before burning up in the planet’s atmosphere.  Astronomers believe that Jupiter was the first planet to form in the Solar System, and so it must have played a critical role in the formation of the remaining planets.   Juno’s mission is to learn how Jupiter was formed, to help us understand how Earth came to be.

Shortly before Juno‘s orbit insertion, Dr. Philip Blanco of Grossmont College mined NASA’s press releases and other on-line tools for solid numerical data on the spacecraft’s past and planned trajectory, to make the orbit insertion maneuver a “teachable moment” for introductory physics students.  Using his data, he kindly wrote a “guest post” (presented below) for use by physics instructors in their introductory mechanics classes.   I edited it just a bit, but all credit for the real work belongs to Phil.  We hope you will find it useful!

 

Eyes on Juno for Jupiter Orbit Insertion

by Philip Blanco, Grossmont College

The Juno spacecraft has reached Jupiter after an epic journey that began with launch from Cape Canaveral in August, 2011.  At the end of May, 2016, when Jupiter’s gravity dominated its motion, mission planners began preparations for the Jupiter Orbit Insertion (JOI) phase of the mission.  (See this melodramatic NASA video.)  In this post, we’ll see how mission information provided by NASA’s Eyes on the Solar System online application can be used along with some introductory mechanics to calculate the  velocity change required to put the spacecraft into its first “capture” orbit.

NASA provided some dynamical information about JOI in a Press Kit.  However, non-metric units and vague statements about speed and distance appearing in the kit made it frustrating for this educator to extract “hard numbers.”  (An alternate source with more details is Spaceflight 101.)  Fortunately, NASA provides a much better resource for our purposes: JPL’s Eyes on the Solar System uses real mission data to simulate solar system and spacecraft motions, in the past, (predicted) present, and near-future.  Eyes gets its data from SPICE format files produced by the mission teams themselves.  Once downloaded (PC and Mac versions are both available), select the Advanced mode and find your favorite mission to “fly along” with.  After first switching to metric units (under Visual Controls on the right of the screen), I found an option under Cool Tools that allows the user to display relative distance and speed between any two selected objects.  I chose  Jupiter and Juno, and noted that the distance shown in km appears to be from Jupiter’s cloud tops, not its center, so I added Jupiter’s equatorial radius to the distances displayed by the tool.

A.  Juno’s Approach Energy and Speed

I used the following physical characteristics of Jupiter: \large GM=1.267\times 10^{8} \mathrm{\: km^{3}/s^{2}}, and equatorial radius \large R=71492\mathrm{ \; km} (used here because the initial approach and subsequent capture orbits pass over the poles with perijove – the closest distance from Jupiter’s center –  above the equator).

For simplicity I analyzed Juno‘s motion relative to Jupiter starting at noon on July 3, 2016.  Although one could say that the Sun’s gravitational pull can be neglected at this point, it is more accurate to say that we can neglect the difference between Juno‘s and Jupiter’s freefall acceleration toward the Sun, i.e., we can treat the Jovian frame of reference as inertial for calculating Juno‘s subsequent motion relative to the planet.

Approximating Jupiter as a spherical body, and ignoring all other influences, Juno‘s orbital mechanical energy per unit mass ε is conserved during its unpowered approach to perijove, and is expressed as:

\large \varepsilon =\frac{1}{2}v^{2}-\frac{GM}{r}.

(1)

Juno‘s incoming speed and distance relative to Jupiter’s center on 2016 July 3.5 were \large r=1.40\times 10^{6}\mathrm{\; km} and \large v=14.49\mathrm{\; km/s}, for which \large \varepsilon _{in}=14.5\mathrm{\; MJ/kg}.  The fact that \large \varepsilon >0 confirms that Juno was on an (unbound) hyperbolic trajectory which would continue around and away from Jupiter unless ε is corrected to a negative value consistent with a bound elliptical orbit.  (Although this value of \large \varepsilon _{in} looks huge, in fact it is tiny compared to the kinetic energy per unit mass needed for escape from the Jovian surface:  \large \frac{1}{2}v_{esc}^{2}=\sqrt{GM/R}=1.77\mathrm{\; GJ/kg}.)

Rearranging Eqn. 1, Juno‘s uncorrected incoming perijove speed is

\large v_{p,in}=\sqrt{2\varepsilon _{in}+\frac{2GM}{r_{p}}}.

(2)

NASA’s Press Kit and Eyes give different values for the perijove distance \large r_{p}, which I derived from their stated altitudes above the Jovian cloud tops.  Let’s take \large r_{p}=76000\pm 200\mathrm{\; km}, or 1.06 Jupiter radii, for which Eqn. 2 yields \large v_{p,in}=58.02\mathrm{\; km/s}.  Compare this to the local escape speed \large v_{esc}=\sqrt{2GM/r_{p}}=57.74\mathrm{\; km/s}.  To enter a capture orbit, the speed must be reduced by at least 0.26 km/s.

B.  Required Speed at Perijove for Capture Orbit

NASA’s Press Kit states that, after firing its main engine, Juno‘s path will be converted from an unbound hyperbolic trajectory to an elliptical “capture” orbit with period \ T=53.5\boldsymbol{\mathrm{\; Earth\: days}}=4620\mathrm{\; ks}.  What instantaneous change of speed (a.k.a. specific impulse) at perijove is required to achieve this new orbit?

Kepler’s 3rd law states that \large T^{2}/a^{3}=4\pi ^{2}/GM, where a is the orbital semi-major axis.  Solving for a, we obtain \large a=4.09\times 10^{6}\mathrm{\; km}, or about 57.2 Jovian radii.  The orbital energy per unit mass of a body in elliptical orbit is given by (See PPE, Eqn. 10.13)

\large \varepsilon _{capture}=-\frac{GM}{2a},

(3)

which for the values given above yields \large \varepsilon _{capture}=-1.548\times 10^{7}\mathrm{\; J/kg}, negative for the bound orbit.  Substituting \large \varepsilon _{capture} and \large r_{p}=76000\mathrm{\; km} into Eqn. 2 gives the perijove speed for this orbit: \large  v_{p,capture}=57.48\mathrm{\; km/s}, just below the local escape speed given above.

Therefore, Juno‘s required change in speed is

\large \Delta v=v_{p,capture}-v_{p,in}=-0.54\mathrm{\; km/s},

very close to the figure of 541 m/s stated in advance by NASA!  The figure below shows the incoming hyperbolic path and the capture orbit predicted by our simple model.

Figure 2
Figure 2:  Incoming hyperbolic trajectory of Juno (blue solid line, projected as blue dashed line) and elliptical capture orbit (green) for values given in text.  (a) Full capture orbit. (b) Close-up of region around perijove.  The distance scale is numbered in units of Jupiter’s equatorial radius.

 

Questions for students

Instructors might ask students to perform some or all of the calculations given above.   Here are a few additional questions related to the Juno mission.

1.  Juno‘s main engine is a British-built  Leros 1b  which burns a mixture of hydrazine (N2H4) and nitrogen tetroxide (N2O4).  A rocket engine’s efficiency is characterized by its specific impulse \large I_{sp}, which is (different from the earlier use of this term) the ratio between the rocket thrust (see PPE, section 6.14) and the weight (on Earth) of fuel consumed per second:

\large F_{thrust}=I_{sp}g\, \dot{m}=v_{ex}\dot{m}

where \large g=9.8\mathrm{\: m/s^{2}}.  The Leros 1b is advertised as having a nominal thrust of 635 N and a specific impulse of 317 s.

a.  What is the exhaust velocity?  (Ans: 3.1 km/s)

b.  What mass of fuel is consumed per second (\large \dot{m})?   How much fuel was consumed in the 35 minute JOI burn?    (Ans: 0.2 kg/s, 430 kg)

c.  Verify that your answers are consistent with the value of  \large \Delta v calculated above.

2.  Following two 53.5 day capture orbits, Juno will execute a Period Reduction Maneuver at perijove to enter its “science” orbit of period 13.965 days = 1206.6 ks.  Assuming that this maneuver does not change the perijove distance, calculate the required \large \Delta v.  (Ans: 400 m/s.  This answer is higher than NASA’s “published” value of 350 m/s.  But the uncertainty in the perijove distance (± 200 km) propagates to an uncertainty in the perijove speed of nearly 100 m/s.  Students might be asked to calculate this uncertainty.)

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